On the determination of solutions of simultaneous Pell equations $$x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1$$ x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

Periodica Mathematica Hungarica - In this paper, we consider the simultaneous Pell equations 0.1 $$\begin{aligned} x^{2}-(a^{2}-1)y^{2}= & {} 1, \nonumber \\ y^{2}-pz^{2}= & {} 1,...     

Genus formula for modular curves of {\mathcal{D}}-elliptic sheaves

We prove a genus formula for modular curves of -elliptic sheaves. We use this formula to show that the reductions of modular curves of -elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of tends to infinity. Received: 14 October 2008 The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Nonholonomic simple \mathcal{D} -Modules over projective varieties

Let X be a smooth complex projective variety whose Picard group is isomorphic to . We prove that there exist simple nonholonomic -modules of holonomic defect 1. Received: 16 March 2005     

A {\mathsf{D}}-induced duality and its applications

This paper attempts to extend the notion of duality for convex cones, by basing it on a prescribed conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the nonnegativity of the inner-product is replaced by a pre-specified conic ordering, defined by a convex cone , and the inner-product itself is replaced by a general multi-dimensional bilinear mapping. This new type of duality is termed the -induced duality in the paper. We further introduce the notion of -induced polar sets within the same framework, which can be viewed as a generalization of the -induced dual cones and is convenient to use for some practical applications. Properties of the extended duality, including the extended bi-polar theorem, are proven. Furthermore, attention is paid to the computation and approximation of the -induced dual objects. We discuss, as examples, applications of the newly introduced -induced duality concepts in robust conic optimization and the duality theory for multi-objective conic optimization. Research supported in part by the Foundation ‘Vereniging Trustfonds Erasmus Universiteit Rotterdam’ in The Netherlands, and in part by Hong Kong RGC Earmarked Grants CUHK4174/03E and CUHK418406.     

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

ON THE DIOPHANTINE EQUATION $\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$

In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.     

A note on the Lebesgue–Ljunggren–Nagell equation $$ax^2+b^{2m}=4y^n$$ a x 2 + b 2 m = 4 y n

Periodica Mathematica Hungarica - Let a, b be fixed positive integers such that $$2\not \mid {ab}$$ , $$\gcd (a,b)=1$$ and a is square-free, and let $$h(-a)$$ denote the class number of...     

Indifferentiable hashing to ordinary elliptic $${\mathbb {F}}_{\!q}$$ F q -curves of $$j=0$$ j = 0 with the cost of one exponentiation in $${\mathbb {F}}_{\!q}$$ F q

Designs, Codes and Cryptography - Let $${\mathbb {F}}_{\!q}$$ be a finite field and $$E_b\!: y^2 = x^3 + b$$ be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that...     

The positive energy conjecture for a class of AHM metrics on R2× Tn-2

We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM) metrics on R²× T^n-2. This generalizes the previous results of Barzegar et al.(2020) as well as Liang and Zhang(2020).     

On the Exponential Diophantine Equation $$(a^{n}-2)(b^{n}-2)=x^{2}$$

Mathematical Notes - In this paper, we deal with the equation $$(a^{n}-2)(b^{n}-2)=x^{2}$$ , $$2\leq a3$$ is odd and $$P_{k},Q_{k}$$ are the Pell and Pell Lucas numbers, respectively. We also...     

Formal structure of direct image of holonomic $${\mathcal{D}}$$ -modules of exponential type

We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic -modules of exponential type. We also prove that every formal -modules is isomorphic, after a ramification, to a germ of formalized direct image of analytic -module of exponential type.     

A note on the diophantine equationx 2+ 4D=y p

LetD be a positive square free integer, and leth(–D) denote the class number of . Furthermore letp be an odd prime with . In this note we prove that ifp {5, 7} orp>3·10⁶, then the equation , has no positive integer solution (x, y).     

$$\tilde{A}$$ and $$\tilde{D}$$ type cluster algebras: triangulated surfaces and friezes

Journal of Algebraic Combinatorics - By viewing $$\tilde{A}$$ and $$\tilde{D}$$ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze...     

Estimates of the Values of $$n$$ -Widths of Classes of Analytic Functions in the Weight Spaces $$H_{2,\gamma}(D)$$

Mathematical Notes - In a simply connected bounded domain $$D\subset\mathbb C$$ with rectifiable Jordan boundary $$\partial D$$ , we study the classes $$H_{2,\gamma}(D;\Omega_k,\Phi)$$ ,...     

关于同伦元素 $\alpha_{1}\beta_{1}\beta_{2}\gamma_{s}$

设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s 相似文献   

Class number one problem for the real quadratic fields $${{\mathbb {Q}({\sqrt{m^2+2r}})}}$$ Q ( m 2 + 2 r )

Archiv der Mathematik - We investigate the class number one problem for a parametric family of real quadratic fields of the form $$\mathbb {Q}( \sqrt{m^2+4r})$$ for certain positive integers m and r.     

A new extension theorem for 3-weight modulo q linear codes over $${\mathbb{F}_ q}$$

We prove that every [n, k, d] _q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .      

On solutions of the simultaneous Pell equations $$ x^{2}-\left( a^{2}-1\right) y^{2}=1$$ and $$y^{2}-pz^{2}=1$$y2-pz2=1

Let \(a\ge 2\) be an integer and p prime number. It is well-known that the solutions of the Pell equation have recurrence relations. For the simultaneous Pell equations

$$\begin{aligned}&x^{2}-\left( a^{2}-1\right) y^{2} =1 \\&y^{2}-pz^{2} =1 \end{aligned}$$

assume that \(x=x_{m}\) and \(y=y_{m}\). In this paper, we show that if \(m\ge 3\) is an odd integer, then there is no positive solution to the system. Moreover, we find the solutions completely for \(5\le a\le 14\) in the cases when \(m\ge 2\) is even integer and \(m=1\).

     

The $${\mathbb {Q}}$$ Q -Korselt set of $$\mathrm {pq}$$ pq

Periodica Mathematica Hungarica - Let N be a positive integer, $${\mathbb {A}}$$ be a nonempty subset of $${\mathbb {Q}}$$ and $$\alpha =\dfrac{\alpha _{1}}{\alpha _{2}}\in {\mathbb {A}}{\setminus...